Skill: decimal-operations
Mastering Decimal Operations
Decimals are just fractions wearing a different outfit — 0.37 means 37/100. Learn when to line up the points, when to count places, and when to slide the point.
Big idea 1 — Place value decides every digit’s job
In 3.47, the 4 is 4 tenths and the 7 is 7 hundredths — the column decides the value, not the digit alone. That column logic is the whole reason lining up the decimal points works: points lined up means tenths add to tenths and hundredths to hundredths.
Place-value chart for 3.47. The decimal point (gold) is the anchor — everything lines up from it.
Big idea 2 — For + and −, line up the points
Adding and subtracting decimals works exactly like whole numbers, as long as you line up the decimal points. Pad with zeros so both numbers show the same decimal places, then add or subtract column by column. The point in the answer drops straight down — it never moves.
0.37 + 0.47 = 0.84. Hundredths combine with hundredths — the picture shows why the decimal point never moves during addition.
Big idea 3 — × counts places, ÷ slides the point
Multiplication: multiply as if the points weren’t there, then place the point by counting total decimal places. 2.4 × 0.35 → 24 × 35 = 840, and 1 + 2 = 3 places → 0.840 = 0.84.
Division: slide the point to make the divisor whole first — move it the same distance in the dividend. 7.5 ÷ 0.25 → slide two places → 750 ÷ 25 = 30.
Estimation is your safety net. Before you trust an answer, round and check: 2.4 × 0.35 ≈ 2 × 0.4 = 0.8, so 0.84 is sensible and 8.4 is not.
Worked examples
Example 1 — Adding: 4.25 + 1.8
Thinking: Line up the decimal points and pad with a zero so both numbers show hundredths: 4.25 + 1.80.
- Write 4.25 + 1.80 — the points are lined up, hundredths under hundredths.
- Add column by column: 5 + 0 = 5 hundredths; 2 + 8 = 10 tenths — write 0, carry 1; 4 + 1 + 1 = 6 ones.
- The point drops straight down: 6.05.
- Check: 4 + 2 = 6, and 0.25 + 0.80 ≈ 1.05, so 6.05 is in the right neighborhood.
Example 2 — Subtracting: 9.6 − 3.47
Thinking: Write 9.6 as 9.60, then subtract column by column, borrowing as needed.
- Pad: 9.60 − 3.47. Points lined up.
- Hundredths: 0 − 7 needs a borrow → 10 − 7 = 3.
- Tenths: 5 (after lending) − 4 = 1. Ones: 9 (after lending) − 3 = 6.
- Answer: 6.13. Check: 9.6 − 3.5 would be 6.1, and we subtracted slightly less than 3.5, so 6.13 makes sense.
Example 3 — Muling: 2.4 × 0.35
Thinking: Ignore the points: 24 × 35 = 840. Count decimal places: 1 + 2 = 3, so place the point to make three decimal digits.
- Whole-number multiplication: 24 × 35 = 840.
- Count places: 2.4 has 1, 0.35 has 2 → 3 total.
- Place the point: 840 → 0.840 = 0.84.
- Check: 2.4 × 0.35 ≈ 2 × 0.4 = 0.8. Estimation catches point-placement errors instantly.
Example 4 — Diving: 7.5 ÷ 0.25
Thinking: Move both points right until the divisor is whole: 7.5 ÷ 0.25 becomes 750 ÷ 25.
- The divisor 0.25 needs 2 slides to become 25.
- Slide the dividend the same distance: 7.5 → 750.
- Whole-number division: 750 ÷ 25 = 30.
- Check: 30 × 0.25 = 7.5.
Common mistakes
Lining up the right edges instead of the decimal points.
Wrong: 4.25 + 1.8 computed as 425 + 18 = 443 → “4.43”.
Right: 4.25 + 1.80 = 6.05. The point is the anchor, not the right edge. Pad with zeros until both numbers show the same decimal places.
Forgetting to count decimal places when multiplying.
Wrong: 2.4 × 0.35 = “84” or “8.4”.
Right: 0.84 (1 + 2 = 3 decimal places). Memory hook: estimate first — 2 × 0.4 ≈ 0.8 — then the point has only one sensible home.
Dividing without clearing the divisor’s decimals.
Wrong: 7.5 ÷ 0.25 treated as 7.5 ÷ 25 = 0.3.
Right: Slide both points → 750 ÷ 25 = 30. Fix: “make the divisor whole first” — move the point in the divisor, then move it the same distance in the dividend.
Quick check
Compute 3.6 + 2.45.
Line up the points: 3.60 + 2.45. Add hundredths (60 + 45 = 105 → write 05, carry 1), then tenths and ones: 6.05.
Compute 0.75 × 40. Where does the point land?
75 × 40 = 3000; 2 decimal places → 30. (0.75 × 40 ≈ 1 × 40 = 40, so 30 is sensible.)
Compute 6.4 ÷ 0.8.
Slide one place: 64 ÷ 8 = 8. Check: 8 × 0.8 = 6.4.